NHS 2020: Forecasting multiple seasonality

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.title[fable]
.sticker-float[![fable](resources/fable.svg)]

## Forecasting with multiple seasonality

.bottom[
### Mitchell O'Hara-Wild (<svg style="height:0.8em;top:.04em;position:relative;fill:#1da1f2;" viewBox="0 0 512 512"><path d="M459.37 151.716c.325 4.548.325 9.097.325 13.645 0 138.72-105.583 298.558-298.558 298.558-59.452 0-114.68-17.219-161.137-47.106 8.447.974 16.568 1.299 25.34 1.299 49.055 0 94.213-16.568 130.274-44.832-46.132-.975-84.792-31.188-98.112-72.772 6.498.974 12.995 1.624 19.818 1.624 9.421 0 18.843-1.3 27.614-3.573-48.081-9.747-84.143-51.98-84.143-102.985v-1.299c13.969 7.797 30.214 12.67 47.431 13.319-28.264-18.843-46.781-51.005-46.781-87.391 0-19.492 5.197-37.36 14.294-52.954 51.655 63.675 129.3 105.258 216.365 109.807-1.624-7.797-2.599-15.918-2.599-24.04 0-57.828 46.782-104.934 104.934-104.934 30.213 0 57.502 12.67 76.67 33.137 23.715-4.548 46.456-13.32 66.599-25.34-7.798 24.366-24.366 44.833-46.132 57.827 21.117-2.273 41.584-8.122 60.426-16.243-14.292 20.791-32.161 39.308-52.628 54.253z"/></svg>[@mitchoharawild](https://twitter.com/mitchoharawild))
### 25 November 2020
### Slides @ [slides.mitchelloharawild.com/nhs2020](https://slides.mitchelloharawild.com/nhs2020)
]

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.title[Hello!]

---
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## Understanding multiple seasonality

---
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.center[
## What is seasonality?
]

> A consistent pattern which repeats over a fixed period of time.

.box-12.opaque[
<img src="figure/ped-seasonality-1.svg" style="display: block; margin: auto;" />
]

---
background-image: linear-gradient(to right, rgba(255, 255, 255, .1), rgba(150, 150, 150, .1)), url("resources/season.jpg")
background-size: cover

.center[
## What is *multiple* seasonality?
]

> Two or more seasonal patterns.

.box-12.opaque[
<img src="figure/ped-multi-seasonality-1.svg" style="display: block; margin: auto;" />
]

---
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.center[
## How can seasonal patterns be modelled?
]

<hr>

<br><br>
.center[
### Seasonal dummy variables

### Lagged terms

### Fourier terms

### Exogenous regressors
]

---

## Modelling seasonal patterns with...

## Seasonal dummy variables

.pull-left[
Produces distinct intercepts for each discrete season.

✅ Simplest to interpret and produce.

❌ Requires many terms for longer periods.

❌ Poorly handles complex seasonality.
]

.pull-right[
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      <td class="gt_row gt_left">0</td>
      <td class="gt_row gt_left">0</td>
      <td class="gt_row gt_left">0</td>
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      <td class="gt_row gt_left">1</td>
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      <td class="gt_row gt_left">0</td>
      <td class="gt_row gt_left">0</td>
      <td class="gt_row gt_left">0</td>
      <td class="gt_row gt_left">0</td>
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      <td class="gt_row gt_left gt_stub">...</td>
      <td class="gt_row gt_left">...</td>
      <td class="gt_row gt_left">...</td>
      <td class="gt_row gt_left">...</td>
      <td class="gt_row gt_left">...</td>
      <td class="gt_row gt_left">...</td>
      <td class="gt_row gt_left">...</td>
    </tr>
  </tbody>
  
  
</table></div>
]

---

## Modelling seasonal patterns with...

## Lagged terms

.pull-left[
*'Today will be similar to yesterday.'*

✅ Very difficult to beat accuracy of.

❌ Limited forecast horizon.

❌ Cannot handle non-integer seasonality.
]

.pull-right[
<img src="figure/lag-plot-1.svg" style="display: block; margin: auto;" />
]

---

## Modelling seasonal patterns with...

## Fourier terms

`$$x_{1,t} = \sin\left(\textstyle\frac{2\pi t}{m}\right), x_{2,t} = \cos\left(\textstyle\frac{2\pi t}{m}\right),$$`

`$$x_{3,t} = \sin\left(\textstyle\frac{4\pi t}{m}\right), x_{4,t} = \cos\left(\textstyle\frac{4\pi t}{m}\right), \ldots$$`
<img src="figure/fourier-plot-1.svg" style="display: block; margin: auto;" />

---

## Modelling seasonal patterns with...

## Fourier terms

.pull-left[
Captures seasonal distribution with a continuous periodic functions.

✅ Handles non-integer seasonality (days in year, weeks in month).

✅ Flexibility of seasonal pattern controllable by number of harmonics.

❌ More complicated to interpret.
]

.pull-right[
<img src="figure/fourier-plot-2-1.svg" style="display: block; margin: auto;" />
]

---

## Modelling seasonal patterns with...

## Exogenous regressors

.pull-left[
Capture seasonality by providing underlying factors like temperature, and events.

✅ Better handling of non-seasonal anomalies.

❌ Data may be hard to find.

❌ Future values must be forecasted.
]

.pull-right[
<img src="figure/exog-plot-1.svg" style="display: block; margin: auto;" />
]
---
class: inverse
background-image: linear-gradient(to right, rgba(50, 50, 50, .5), rgba(50, 50, 50, .4)), url("resources/tree.jpg")
background-size: cover

.center[
## Which seasonal method should I use?
]

<hr>

<br>

.center[

### For *multiple* seasonality, *multiple* methods may be needed.
<br>

### Use methods that are available and work best for your data!
]

---

## Models for forecasting multiple seasonalities

Capability for including seasonal terms.

<table class="table">
  <thead>
    <tr>
      <th>Model</th>
      <th>Dummies</th>
      <th>Lags</th>
      <th>Fourier</th>
      <th>Regressors</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>MLR</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
    </tr>
    <tr>
      <td>GAM</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
    </tr>
    <tr>
      <td>Prophet</td>
      <td class="danger">No</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
    </tr>
    <tr>
      <td>DSHW</td>
      <td class="success">Yes</td>
      <td class="danger">No</td>
      <td class="danger">No</td>
      <td class="danger">No</td>
    </tr>
    <tr>
      <td>BATS/TBATS</td>
      <td class="success">Yes</td>
      <td class="warning">Partial</td>
      <td class="success">Yes</td>
      <td class="danger">No</td>
    </tr>
    <tr>
      <td>FASSTER</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
    </tr>
    <tr>
      <td>STL + ???</td>
      <td class="warning">???</td>
      <td class="warning">???</td>
      <td class="warning">???</td>
      <td class="warning">???</td>
    </tr>
  </tbody>
</table>

.footnote[(This *non-exhaustive* list includes the most common statistical methods only.)]

---

## Models for forecasting multiple seasonalities

<br>
<table class="table">
  <thead>
    <tr>
      <th>Model</th>
      <th>Flexible</th>
      <th>Speed</th>
      <th>Accuracy</th>
      <th>Decompose Components</th>
      <th>Evolving Terms</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Regression<br>(MLR, GAM, Prophet)</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>State&nbspSpace<br>(DSHW, TBATS, FASSTER)</td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
    <tr>
      <td>Decomposition<br>(STL + ???)</td>
      <td></td>
      <td>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</td>
      <td></td>
      <td></td>
      <td></td>
    </tr>
  </tbody>
</table>

---

## Models for forecasting multiple seasonalities

<br>
<table class="table">
  <thead>
    <tr>
      <th>Model</th>
      <th>Flexible</th>
      <th>Speed</th>
      <th>Accuracy</th>
      <th>Decompose Components</th>
      <th>Evolving Terms</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Regression<br>(MLR, GAM, Prophet)</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
    </tr>
    <tr>
      <td>State&nbspSpace<br>(DSHW, TBATS, FASSTER)</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
    </tr>
    <tr>
      <td>Decomposition<br>(STL + ???)</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends&nbsp;&nbsp;</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
      <td class="warning">Depends</td>
    </tr>
  </tbody>
</table>

---

## Models for forecasting multiple seasonalities

<br>
<table class="table">
  <thead>
    <tr>
      <th>Model</th>
      <th>Flexible</th>
      <th>Speed</th>
      <th>Accuracy</th>
      <th>Decompose Components</th>
      <th>Evolving Terms</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Regression<br>(MLR, GAM, Prophet)</td>
      <td class="success">Yes</td>
      <td class="success">Fast</td>
      <td class="warning">Okay</td>
      <td class="warning">Depends</td>
      <td class="danger">No</td>
    </tr>
    <tr>
      <td>State&nbspSpace<br>(DSHW, TBATS, FASSTER)</td>
      <td class="danger">No</td>
      <td class="danger">Slow</td>
      <td class="warning">Okay</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
    </tr>
    <tr>
      <td>Decomposition<br>(STL + ???)</td>
      <td class="warning">Depends</td>
      <td class="warning">Moderate</td>
      <td class="warning">Okay</td>
      <td class="success">Yes</td>
      <td class="warning">Depends</td>
    </tr>
  </tbody>
</table>

.footnote[(This table is a *massive* generalisation, everything depends on data and model complexity.)]

---
class: center

## Forecasting multiple seasonality in R

.sticker[![fable](resources/fable.svg)]

.animated.fadeIn[
.sticker[![tsibble](resources/tsibble.svg)]
.sticker[![tsibbledata](resources/tsibbledata.svg)]
.sticker[![feasts](resources/feasts.svg)]

## [tidyverts.org](http://www.tidyverts.org)
]

---
class: top

---
class: inverse, center
background-image: linear-gradient(to right, rgba(50, 50, 50, .5), rgba(50, 50, 50, .5)), url("resources/disk.jpg")
background-size: cover

.title[Tidy]

## Data preparation

---
class: top

# Hospital admissions

```r
fs::dir_tree("~/github/webinar-msc-nhs/")
```

```
#> ~/github/webinar-msc-nhs/
#> ├── README.md
#> ├── data
#> │   ├── ae_uk.csv
#> │   ├── holiday.csv
#> │   └── temp.csv
#> ├── rscript.R
#> └── webinar-msc-nhs.Rproj
```

.footnote[*https://github.com/bahmanrostamitabar/webinar-msc-nhs/*]

---
class: top

# Admissions event data

```r
admissions <- read_csv(
  file = "~/github/webinar-msc-nhs/data/ae_uk.csv",
  col_types = cols(arrival_time=col_datetime(format = "%d/%m/%Y %H:%M"))
) %>%
  select(-ID) # ID is a redundant row identifier 
print(admissions)
```

```
#> # A tibble: 773,779 x 3
#>    gender type_injury arrival_time       
#>    <chr>  <chr>       <dttm>             
#>  1 male   minor       2010-01-01 00:35:00
#>  2 male   minor       2010-01-01 00:57:00
#>  3 male   minor       2010-01-01 00:42:00
#>  4 male   minor       2010-01-01 00:02:00
#>  5 male   minor       2010-01-01 00:24:00
#>  6 female major       2010-01-01 00:48:00
#>  7 female minor       2010-01-01 00:10:00
#>  8 male   minor       2010-01-01 00:39:00
#>  9 female minor       2010-01-01 00:56:00
#> 10 female minor       2010-01-01 00:05:00
#> # … with 773,769 more rows
```

---
class: top

# Holiday data

```r
holidays <- read_csv(
  file = "~/github/webinar-msc-nhs/data/holiday.csv",
  col_types = cols(date=col_date(format = "%d/%m/%Y"))
) %>% 
  filter(!is.na(date)) # Some extra empty rows exist that need removing
print(holidays)
```

```
#> # A tibble: 2,282 x 7
#>    date       public_holiday_type public_holiday school_holiday school_holiday_type  celebratory_day celebratory_day_ty…
#>    <date>     <chr>                        <dbl>          <dbl> <chr>                          <dbl> <chr>              
#>  1 2010-01-01 New Years Day                    1              0 0                                  0 0                  
#>  2 2010-01-02 0                                0              1 Winter School Holid…               0 0                  
#>  3 2010-01-03 0                                0              1 Winter School Holid…               0 0                  
#>  4 2010-01-04 0                                0              0 0                                  0 0                  
#>  5 2010-01-05 0                                0              0 0                                  0 0                  
#>  6 2010-01-06 0                                0              0 0                                  0 0                  
#>  7 2010-01-07 0                                0              0 0                                  0 0                  
#>  8 2010-01-08 0                                0              0 0                                  0 0                  
#>  9 2010-01-09 0                                0              0 0                                  0 0                  
#> 10 2010-01-10 0                                0              0 0                                  0 0                  
#> # … with 2,272 more rows
```

---
class: top

# Temperature data

```r
temperatures <- read_csv(
  file = "~/github/webinar-msc-nhs/data/temp.csv",
  col_types = cols(date=col_date(format = "%d/%m/%Y"))
)

print(temperatures)
```

```
#> # A tibble: 2,282 x 3
#>    date       actual_temp forecast_temp
#>    <date>           <dbl>         <dbl>
#>  1 2010-01-01         4             4.3
#>  2 2010-01-02         4.8           5  
#>  3 2010-01-03         6             6.4
#>  4 2010-01-04         3             1.9
#>  5 2010-01-05         4.1           3.8
#>  6 2010-01-06         3.5           2.2
#>  7 2010-01-07         1.6           1  
#>  8 2010-01-08         3             2.7
#>  9 2010-01-09         1.5           1  
#> 10 2010-01-10         1.4           1.3
#> # … with 2,272 more rows
```

---

class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Tidy temporal data structures

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Tidy temperature data

Identify the index, key and measurement variables:

```r
temperatures <- as_tsibble(temperatures, 
*          index = ???, key = ???)
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Tidy temperature data

Identify the index, key and measurement variables:

```r
temperatures <- as_tsibble(temperatures, 
*          index = date, key = NULL)
```

```
#> # A tsibble: 2,282 x 3 [1D]
#>    date       actual_temp forecast_temp
#>    <date>           <dbl>         <dbl>
#>  1 2010-01-01         4             4.3
#>  2 2010-01-02         4.8           5  
#>  3 2010-01-03         6             6.4
#>  4 2010-01-04         3             1.9
#>  5 2010-01-05         4.1           3.8
#>  6 2010-01-06         3.5           2.2
#>  7 2010-01-07         1.6           1  
#>  8 2010-01-08         3             2.7
#>  9 2010-01-09         1.5           1  
#> 10 2010-01-10         1.4           1.3
#> # … with 2,272 more rows
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Tidy holiday data

Identify the index, key and measurement variables:

```r
holidays <- as_tsibble(holidays, 
*          index = ???, key = ???)
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Tidy holiday data

Identify the index, key and measurement variables:

```r
holidays <- as_tsibble(holidays, 
*          index = date, key = NULL)
```

```
#> # A tsibble: 2,282 x 7 [1D]
#>    date       public_holiday_type public_holiday school_holiday school_holiday_type  celebratory_day celebratory_day_ty…
#>    <date>     <chr>                        <dbl>          <dbl> <chr>                          <dbl> <chr>              
#>  1 2010-01-01 New Years Day                    1              0 0                                  0 0                  
#>  2 2010-01-02 0                                0              1 Winter School Holid…               0 0                  
#>  3 2010-01-03 0                                0              1 Winter School Holid…               0 0                  
#>  4 2010-01-04 0                                0              0 0                                  0 0                  
#>  5 2010-01-05 0                                0              0 0                                  0 0                  
#>  6 2010-01-06 0                                0              0 0                                  0 0                  
#>  7 2010-01-07 0                                0              0 0                                  0 0                  
#>  8 2010-01-08 0                                0              0 0                                  0 0                  
#>  9 2010-01-09 0                                0              0 0                                  0 0                  
#> 10 2010-01-10 0                                0              0 0                                  0 0                  
#> # … with 2,272 more rows
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Tidy admissions data

Identify the index, key and measurement variables:

```r
admissions <- as_tsibble(admissions, 
*          index = ???, key = ???)
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Tidy admissions data

Identify the index, key and measurement variables:

```r
admissions <- as_tsibble(admissions, 
*          index = arrival_time, key = c(gender, type_injury))
```

```
#> Error: A valid tsibble must have distinct rows identified by key and index.
#> ℹ Please use `duplicates()` to check the duplicated rows.
```

```r
duplicates(admissions, index = arrival_time, key = c(gender, type_injury))
```

```
#> # A tibble: 82,578 x 3
#>    gender type_injury arrival_time       
#>    <chr>  <chr>       <dttm>             
#>  1 female minor       2010-01-01 01:37:00
#>  2 female minor       2010-01-01 01:37:00
#>  3 female minor       2010-01-01 10:07:00
#>  4 female minor       2010-01-01 10:07:00
#>  5 female minor       2010-01-01 11:15:00
#>  6 female minor       2010-01-01 11:15:00
#>  7 male   minor       2010-01-01 12:01:00
#>  8 female minor       2010-01-01 12:24:00
#>  9 female minor       2010-01-01 12:24:00
#> 10 male   minor       2010-01-01 12:01:00
#> # … with 82,568 more rows
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Tidy admissions data

Identify the index, key and measurement variables:

```r
admissions <- admissions %>% 
* count(gender, type_injury, arrival_time, name = "arrivals") %>%
  as_tsibble(index = arrival_time, key = c(gender, type_injury))
```

```
#> # A tsibble: 731,906 x 4 [1m] <UTC>
#> # Key:       gender, type_injury [4]
#>    gender type_injury arrival_time        arrivals
#>    <chr>  <chr>       <dttm>                 <int>
#>  1 female major       2010-01-01 00:48:00        1
#>  2 female major       2010-01-01 04:19:00        1
#>  3 female major       2010-01-01 05:57:00        1
#>  4 female major       2010-01-01 12:17:00        1
#>  5 female major       2010-01-01 12:42:00        1
#>  6 female major       2010-01-01 13:34:00        1
#>  7 female major       2010-01-01 16:59:00        1
#>  8 female major       2010-01-02 23:56:00        1
#>  9 female major       2010-01-03 00:24:00        1
#> 10 female major       2010-01-03 13:14:00        1
#> # … with 731,896 more rows
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Bringing the data together

Can be a non-trivial task!

```r
interval(admissions)
```

```
#> <interval[1]>
#> [1] 1m
```

```r
interval(holidays)
```

```
#> <interval[1]>
#> [1] 1D
```

```r
interval(temperatures)
```

```
#> <interval[1]>
#> [1] 1D
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Choosing appropriate temporal intervals

What temporal granularity should `admissions` have?

A balancing act between:

* The temporal detail needed for the forecasts
* Enough detail/signal in the data for a good model

This choice has a big impact on the seasonal structures that will exist in the data!

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Choosing appropriate temporal intervals

Option A: Keep it as is (event level)

```r
admissions %>% 
  filter(yearmonth(arrival_time) == yearmonth("2010 Jan")) %>% 
* summarise(arrivals = sum(arrivals)) %>%
* fill_gaps(arrivals = 0) %>%
  autoplot()
#
#
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Choosing appropriate temporal intervals

Option B: 30 minutes

```r
admissions %>% 
  filter(yearmonth(arrival_time) == yearmonth("2010 Jan")) %>% 
* index_by(time = floor_date(arrival_time, "30 minutes")) %>%
  summarise(arrivals = sum(arrivals)) %>% 
  fill_gaps(arrivals = 0) %>% 
  autoplot()
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Choosing appropriate temporal intervals

Option C: 1 hour

```r
admissions %>% 
  filter(yearmonth(arrival_time) == yearmonth("2010 Jan")) %>% 
* index_by(time = floor_date(arrival_time, "1 hour")) %>%
  summarise(arrivals = sum(arrivals)) %>% 
  fill_gaps(arrivals = 0) %>% 
  autoplot()
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Choosing appropriate temporal intervals

Option D: 1 day

```r
admissions %>% 
  filter(year(arrival_time) == 2010) %>% 
* index_by(time = as.Date(floor_date(arrival_time, "1 day"))) %>%
  summarise(arrivals = sum(arrivals)) %>% 
  fill_gaps(arrivals = 0) %>% 
  autoplot()
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Choosing appropriate temporal intervals

Option E: 1 month

```r
admissions %>% 
* index_by(time = yearmonth(floor_date(arrival_time, "1 day"))) %>%
  summarise(arrivals = sum(arrivals)) %>% 
  fill_gaps(arrivals = 0) %>%
  autoplot()
#
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Choosing appropriate temporal intervals

So what's best?

> * Has the temporal detail needed for the forecasts
> * Enough detail/signal in the data for a good model

.center[*It depends!*]

.center[
# 🙄
]

.center[For a multiple seasonality webinar, probably B or C.

Let's go with hourly data (C) for as it has better signal.]

---

class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Choosing appropriate temporal intervals

```r
admissions <- admissions %>% 
* index_by(time = floor_date(arrival_time, "1 hour")) %>%
  summarise(arrivals = sum(arrivals)) %>% 
  ungroup() %>% 
  fill_gaps(arrivals = 0)
```

```
#> # A tsibble: 54,768 x 2 [1h] <UTC>
#>    time                arrivals
#>    <dttm>                 <dbl>
#>  1 2010-01-01 00:00:00       13
#>  2 2010-01-01 01:00:00       15
#>  3 2010-01-01 02:00:00       12
#>  4 2010-01-01 03:00:00       15
#>  5 2010-01-01 04:00:00       11
#>  6 2010-01-01 05:00:00        7
#>  7 2010-01-01 06:00:00        6
#>  8 2010-01-01 07:00:00        4
#>  9 2010-01-01 08:00:00        6
#> 10 2010-01-01 09:00:00       10
#> # … with 54,758 more rows
```

---
class: middle
background-image: linear-gradient(to right, rgba(50, 50, 50, .5), rgba(50, 50, 50, .5)), url("resources/many_clocks.jpg")
background-size: cover

.box-12[
## Aside

Advanced forecasters may even choose multiple granularities!

<br>

Forecasts at many levels can be combined using **temporal reconciliation** to
get more accurate results.

This process ensures the hourly forecasts add to the
daily forecasts, and so on.

<br>

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Bringing the data together

```r
interval(admissions)
```

```
#> <interval[1]>
#> [1] 1h
```

```r
interval(holidays)
```

```
#> <interval[1]>
#> [1] 1D
```

```r
interval(temperatures)
```

```
#> <interval[1]>
#> [1] 1D
```

It's okay for `holidays` to be daily as the holiday affects the entire day.

Ideally, `temperatures` would be hourly as it varies throughout the day.

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Bringing the data together

```r
admissions %>% 
  # Compute the common variable
  mutate(date = as.Date(time))
```

```
#> # A tsibble: 54,768 x 3 [1h] <UTC>
#>    time                arrivals date      
#>    <dttm>                 <dbl> <date>    
#>  1 2010-01-01 00:00:00       13 2010-01-01
#>  2 2010-01-01 01:00:00       15 2010-01-01
#>  3 2010-01-01 02:00:00       12 2010-01-01
#>  4 2010-01-01 03:00:00       15 2010-01-01
#>  5 2010-01-01 04:00:00       11 2010-01-01
#>  6 2010-01-01 05:00:00        7 2010-01-01
#>  7 2010-01-01 06:00:00        6 2010-01-01
#>  8 2010-01-01 07:00:00        4 2010-01-01
#>  9 2010-01-01 08:00:00        6 2010-01-01
#> 10 2010-01-01 09:00:00       10 2010-01-01
#> # … with 54,758 more rows
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Bringing the data together

```r
admissions %>% 
  # Compute the common variable
  mutate(date = as.Date(time)) %>% 
  # Join in holiday data (keeping only 'public_holiday' for simplicity)
  left_join(holidays %>% select(public_holiday), by = "date")
```

```
#> # A tsibble: 54,768 x 4 [1h] <UTC>
#>    time                arrivals date       public_holiday
#>    <dttm>                 <dbl> <date>              <dbl>
#>  1 2010-01-01 00:00:00       13 2010-01-01              1
#>  2 2010-01-01 01:00:00       15 2010-01-01              1
#>  3 2010-01-01 02:00:00       12 2010-01-01              1
#>  4 2010-01-01 03:00:00       15 2010-01-01              1
#>  5 2010-01-01 04:00:00       11 2010-01-01              1
#>  6 2010-01-01 05:00:00        7 2010-01-01              1
#>  7 2010-01-01 06:00:00        6 2010-01-01              1
#>  8 2010-01-01 07:00:00        4 2010-01-01              1
#>  9 2010-01-01 08:00:00        6 2010-01-01              1
#> 10 2010-01-01 09:00:00       10 2010-01-01              1
#> # … with 54,758 more rows
```

---
class: top

.sticker-float[![tsibble](resources/tsibble.svg)]

# Bringing the data together

```r
hospital <- admissions %>% 
  # Compute the common variable
  mutate(date = as.Date(time)) %>% 
  # Join in holiday data (keeping only 'public_holiday' for simplicity)
  left_join(holidays %>% select(public_holiday), by = "date") %>% 
  # Join in temperature data (keeping only 'actual_temp' for simplicity)
  left_join(temperatures %>% select(actual_temp), by = "date")
```

```
#> # A tsibble: 54,768 x 5 [1h] <UTC>
#>    time                arrivals date       public_holiday actual_temp
#>    <dttm>                 <dbl> <date>              <dbl>       <dbl>
#>  1 2010-01-01 00:00:00       13 2010-01-01              1           4
#>  2 2010-01-01 01:00:00       15 2010-01-01              1           4
#>  3 2010-01-01 02:00:00       12 2010-01-01              1           4
#>  4 2010-01-01 03:00:00       15 2010-01-01              1           4
#>  5 2010-01-01 04:00:00       11 2010-01-01              1           4
#>  6 2010-01-01 05:00:00        7 2010-01-01              1           4
#>  7 2010-01-01 06:00:00        6 2010-01-01              1           4
#>  8 2010-01-01 07:00:00        4 2010-01-01              1           4
#>  9 2010-01-01 08:00:00        6 2010-01-01              1           4
#> 10 2010-01-01 09:00:00       10 2010-01-01              1           4
#> # … with 54,758 more rows
```

---
class: inverse, center
background-image: linear-gradient(to right, rgba(50, 50, 50, .5), rgba(50, 50, 50, .5)), url("resources/graph.jpg")
background-size: cover

.title[Visualise]

<hr>

<br>

## Data exploration

---

# Visualising multiple seasonality

Plotting hourly data observed over six years can be tricky!

```r
hospital %>% 
  autoplot(arrivals)
```

---

# Visualising multiple seasonality

One possibility is to view a portion of the data (but you might miss things!)

```r
library(dygraphs)
tsbox::ts_xts(hospital) %>% 
  dygraph() %>% 
  dyRangeSelector(dateWindow = c("2010-01-01", "2010-02-01"))
```

<div id="htmlwidget-650b8c943c6c24ffa67e" style="width:864px;height:288px;" class="dygraphs html-widget"></div>
<script type="application/json" data-for="htmlwidget-650b8c943c6c24ffa67e">{"x":{"attrs":{"labels":["hour","arrivals","public_holiday","actual_temp"],"legend":"auto","retainDateWindow":false,"axes":{"x":{"pixelsPerLabel":60}},"showRangeSelector":true,"dateWindow":["2010-01-01T00:00:00.000Z","2010-02-01T00:00:00.000Z"],"rangeSelectorHeight":40,"rangeSelectorPlotFillColor":" 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---

.sticker-float[![ggplot2](resources/feasts.svg)]

# Visualising multiple seasonality

Seeing the big picture may require aggregating the data.

```r
library(feasts)
hospital %>% 
  index_by(date = yearmonth(time)) %>% 
  summarise(arrivals = sum(arrivals)) %>% 
  gg_subseries(arrivals, period = "year")
```

---

.sticker-float[![ggplot2](resources/feasts.svg)]

# Visualising multiple seasonality

```r
hospital %>% 
  index_by(date = as.Date(time)) %>% 
  summarise(arrivals = sum(arrivals)) %>% 
  gg_subseries(arrivals, period = "week")
```

---

.sticker-float[![ggplot2](resources/feasts.svg)]

# Visualising multiple seasonality

```r
hospital %>% 
  gg_subseries(arrivals, period = "day")
```

---

.sticker-float[![ggplot2](resources/ggplot2.svg)]

# Visualising multiple seasonality

```r
hospital %>% 
  ggplot(aes(x = hour(time), y = arrivals)) + 
  geom_jitter(alpha = 0.01) +
  geom_smooth() + 
  facet_grid(cols = vars(wday(time, label = TRUE, week_start = 1)))
```

---

.sticker-float[![ggplot2](resources/ggplot2.svg)]

# Visualising multiple seasonality

```r
*library(sugrrants)
p <- hospital %>% 
  mutate(date = as.Date(time)) %>% 
  filter(year(time) == 2012) %>% 
* frame_calendar(x = hour(time), y = arrivals, date = date) %>%
  ggplot(aes(x = `.hour(time)`, y = .arrivals, group = date)) + 
  geom_line()
prettify(p)
```

---
.sticker-float[![ggplot2](resources/ggplot2.svg)]

# Visualising multiple seasonality

---
class: inverse, center
background-image: linear-gradient(to right, rgba(50, 50, 50, .7), rgba(50, 50, 50, .7)), url("resources/books.jpg")
background-size: cover

.title[Making forecasts]

<hr>

<br>

## Data modelling

.left[
.footnote[
⚠️ The specific models/forecasts shown are in no way recommended or good.

They simply demonstrate how multiple seasonalities can be modelled.]
]

---

## Models for forecasting multiple seasonalities

<br>
<table class="table">
  <thead>
    <tr>
      <th>Model</th>
      <th>Flexible</th>
      <th>Speed</th>
      <th>Accuracy</th>
      <th>Decompose Components</th>
      <th>Evolving Terms</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Regression<br>(MLR, GAM, <b>Prophet</b>)</td>
      <td class="success">Yes</td>
      <td class="success">Fast</td>
      <td class="warning">Okay</td>
      <td class="warning">Depends</td>
      <td class="danger">No</td>
    </tr>
    <tr>
      <td>State&nbspSpace<br>(DSHW, TBATS, <b>FASSTER</b>)</td>
      <td class="danger">No</td>
      <td class="danger">Slow</td>
      <td class="warning">Okay</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
    </tr>
    <tr>
      <td>Decomposition<br>(<b>STL + ???</b>)</td>
      <td class="warning">Depends</td>
      <td class="warning">Moderate</td>
      <td class="warning">Okay</td>
      <td class="success">Yes</td>
      <td class="warning">Depends</td>
    </tr>
  </tbody>
</table>

.footnote[(This table is a *massive* generalisation, everything depends on data and model complexity.)]

---

## Regression modelling

<br>

<table class="table">
  <thead>
    <tr>
      <th>Model</th>
      <th>Flexible</th>
      <th>Speed</th>
      <th>Accuracy</th>
      <th>Decompose Components</th>
      <th>Evolving Terms</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Regression<br>(MLR, GAM, <b>Prophet</b>)</td>
      <td class="success">Yes</td>
      <td class="success">Fast</td>
      <td class="warning">Okay</td>
      <td class="warning">Depends</td>
      <td class="danger">No</td>
    </tr>
  </tbody>
</table>

Uses time series features / regressors with known future values to produce forecasts.

✅ Generally fast, as model estimation is time independent.

❌ Model behaviour does not update over time for changing dynamics.

Highly recommended read on GAM forecasting: https://petolau.github.io/Analyzing-double-seasonal-time-series-with-GAM-in-R/
---

## Prophet

Facebook's forecasting model based on a GAM which uses regressors for growth, seasonality, and holidays.

$$
  y(t) = g(t) + s_1(t) + s_2(t) + h(t) + \varepsilon_t
$$

where `$g(t)$` is the growth function, `$s(t)$` are fourier terms, and `$h(t)$` are holiday regressors.

Available directly in the [`{prophet}`](https://facebook.github.io/prophet/) package, or a fable interface from [`{fable.prophet}`](http://pkg.mitchelloharawild.com/fable.prophet/).

---

## Specifying a prophet model

Using default settings

```r
library(fable.prophet)
hospital %>% 
  model(prophet(arrivals)) %>% 
  forecast(h = "2 weeks") %>% 
  autoplot(tail(hospital, 24*7*4))
```

---

## Specifying a prophet model

Customising the `season()` options.

```r
library(fable.prophet)
hospital %>% 
  model(prophet(arrivals ~ season("day", 7, type = "mult") + season("week", 4))) %>% 
  forecast(h = "2 weeks") %>% 
  autoplot(tail(hospital, 24*7*4))
```

---

## State space modelling

<br>

<table class="table">
  <thead>
    <tr>
      <th>Model</th>
      <th>Flexible</th>
      <th>Speed</th>
      <th>Accuracy</th>
      <th>Decompose Components</th>
      <th>Evolving Terms</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>State&nbspSpace<br>(DSHW, TBATS, <b>FASSTER</b>)</td>
      <td class="danger">No</td>
      <td class="danger">Slow</td>
      <td class="warning">Okay</td>
      <td class="success">Yes</td>
      <td class="success">Yes</td>
    </tr>
  </tbody>
</table>

Iterates 'state equations' over time to produce forecasts.

✅ Model behaviour updates for changes in the data.

❌ Relatively slow, as model estimation is done sequentially over time.

---

## FASSTER

A model which provides flexible regression-style model specification with evolving terms of state space models via a DLM.

`\begin{align*}
y_t &= F_t\theta_t + v_t, & v_t &\sim \mathcal{N}(0,V)\\
\theta_t &= G\theta_{t-1} + w_t, & w_t&\sim \mathcal{N}(0,W)
\end{align*}`

The underlying states are given by `$\theta$`, which can be thought of as components linearly combined by `$F_t$` to produce the response `$y_t$`. The `$G$` matrix defines the behaviour of each state (seasonality, trend, regression, ...).

Available with direct fable compatibility in the [`{fasster}`](http://fasster.tidyverts.org/) package.

---

## Specifying seasonality with FASSTER

FASSTER allows all aforementioned seasonal terms to be specified.

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<div id="gbhorzsgjc" style="overflow-x:auto;overflow-y:auto;width:auto;height:auto;"><table class="gt_table">
  
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Seasonal term</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">Special</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr>
      <td class="gt_row gt_left">Seasonal dummies</td>
      <td class="gt_row gt_left">y ~ season(p)</td>
    </tr>
    <tr>
      <td class="gt_row gt_left">Lag terms</td>
      <td class="gt_row gt_left">y ~ lag(y, p)</td>
    </tr>
    <tr>
      <td class="gt_row gt_left">Fourier terms</td>
      <td class="gt_row gt_left">y ~ fourier(p, K)</td>
    </tr>
    <tr>
      <td class="gt_row gt_left">Regressors</td>
      <td class="gt_row gt_left">y ~ x</td>
    </tr>
  </tbody>
  
  
</table></div>

---

## Specifying several seasonalities with FASSTER

```r
library(fasster)
ped_train <- pedestrian %>% 
  filter(Sensor == "Southern Cross Station",
         yearmonth(Date) == yearmonth("2015 July"))
ped_train %>% 
  model(fasster(Count ~ trend(1) + fourier("week", 3) + (wday(Date_Time) %in% c(1,7)) %S% fourier("day", 10))) %>% 
  forecast(h = "2 weeks") %>% 
  autoplot(ped_train)
```

---

## Decomposition modelling

<br>

<table class="table">
  <thead>
    <tr>
      <th>Model</th>
      <th>Flexible</th>
      <th>Speed</th>
      <th>Accuracy</th>
      <th>Decompose Components</th>
      <th>Evolving Terms</th>
    </tr>
  </thead>
  <tbody>
    <tr>
      <td>Decomposition<br>(<b>STL + ???</b>)</td>
      <td class="warning">Depends</td>
      <td class="warning">Moderate</td>
      <td class="warning">Okay</td>
      <td class="success">Yes</td>
      <td class="warning">Depends</td>
    </tr>
  </tbody>
</table>

Decomposes time series into components, allowing seasonality to be forecasted
separately with simpler methods.

✅ Allows complex time series to be forecasted with simpler models.

❌ Highly dependent on the ability to decompose the time series.

---

## STL + ???

Uses repeated STL decompositions to decompose multiple seasonalities into separate time series. Then other models (`???`) can be used to forecast the simpler components.

`$$y_{t} = S_{\text{year},t} + S_{\text{week},t} + S_{\text{day},t} + T_{t} + R_t$$`

Available with direct fable compatibility in the [`{feasts}`](http://feasts.tidyverts.org/) package.

---

## Decomposing a time series

Using an `STL` decomposition with default settings:

```r
library(feasts)
hospital_dcmp <- hospital %>% 
  model(STL(arrivals)) 
components(hospital_dcmp) %>% autoplot()
```

---

## Decomposing a time series

Using an `STL` decomposition with default settings:

```r
components(hospital_dcmp) %>% 
  tail(24*7*5) %>% 
  autoplot()
```

---

## Decomposing a time series

Refining the loess windows for the sub-daily data frequency:

```r
hospital_dcmp <- hospital %>% 
  model(STL(arrivals ~ trend(window = 24*7*5) + season("year", window = Inf) +
              season("week", window = 24*7*5) + season("day", window = 24*7)))
components(hospital_dcmp) %>% autoplot()
```

---

## Decomposing a time series

Refining the loess windows for the sub-daily data frequency:

```r
components(hospital_dcmp) %>% 
  tail(24*7*5) %>% 
  autoplot()
```

---

## Modelling decomposed components

As seasonality changes slowly, it is easily modelled with naive lags.

```r
components(hospital_dcmp)[-1] %>% 
  model(SNAIVE(season_week ~ lag("week"))) %>% 
  forecast(h = "2 weeks") %>% 
  autoplot(tail(components(hospital_dcmp)[-1], 24*7*4))
```

---

## Modelling decomposed components

As seasonality changes slowly, it is easily modelled with naive lags.

```r
components(hospital_dcmp)[-1] %>% 
  model(SNAIVE(season_day ~ lag("day"))) %>% 
  forecast(h = "2 weeks") %>% 
  autoplot(tail(components(hospital_dcmp)[-1], 24*7*4))
```

---

## Modelling decomposed components

The seasonally adjusted data (trend + remainder) is typically forecasted together.

```r
components(hospital_dcmp)[-1] %>% 
  model(ARIMA(season_adjust ~ 0 + pdq(3,0,3) + PDQ(1,0,0))) %>% 
  forecast(h = "3 days") %>% 
  autoplot(tail(components(hospital_dcmp)[-1], 24*4))
```

---

## Combining individual forecasts

Use `decomposition_model()` to specify all component models:

```r
dcmp_spec <- decomposition_model(
  dcmp = STL(arrivals ~ trend(window = 24*7*5) + season("year", window = Inf) +
              season("week", window = 24*7*5) + season("day", window = 24*7)),
  ARIMA(season_adjust ~ 0 + pdq(3,0,3) + PDQ(1,0,0)),
  SNAIVE(season_week ~ lag("week")),
  SNAIVE(season_day ~ lag("day"))
)
```

Notice that I haven't provided a model for `season_year`. A `SNAIVE()` model will
be used by default for seasonal components.

---

## Combining individual forecasts

Then use it as a regular model, the recombination is done automatically.

```r
hospital %>% 
  model(dcmp_spec) %>% 
  forecast(h = "2 weeks") %>% 
  autoplot(tail(hospital, 24*7*4))
```

---
class: inverse, top
background-image: linear-gradient(to right, rgba(150, 150, 150, .1), rgba(150, 150, 150, .4)), url("resources/hourglass.jpg")
background-size: cover

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